Etymology
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Words related to irrepressible

in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
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repress (v.)

late 14c., "to check, restrain (sin, error); to overcome, put down, subdue (riot, rebellion);" from Latin repressus, past participle of reprimere "hold back, curb," figuratively "check, confine, restrain, refrain," from re- "back" (see re-) + premere "to press, hold fast, cover, crowd, compress" (from PIE root *per- (4) "to strike").

Used of feelings or desires from late 14c.; in the purely psychological sense "keep out of the conscious mind, keep in the subconscious" it represents German verdrängen (Freud, 1893), that sense of the word first attested in English in 1904 (implied in repressed). Related: Repressed; repressing.

-ible 
word-forming element making adjectives from verbs, borrowed in Middle English from Old French -ible and directly from Latin adjective suffix -ibilis (properly -bilis); see -able.
repressible (adj.)

"capable of being restrained," 1801, of emotions and indignation or of medical conditions, but if the former almost always qualified by hardly, no longer, barely, etc.; see repress (v.) + -ible. Also compare irrepressible.